This implies that \({\hat{P}}u=f\) . Since we may assume that \(2n-1\ge 0\)  and hence \(\langle {D}\rangle ^2u\in L^2((0,\delta )\times {\mathbb {R}}^d)\)  it follows from [1]} that
\(\langle {D}\rangle ^{2-j}D_t^ju\in L^2((0,\delta )\times {\mathbb {R}}^d),\quad j=0,1,2.\) 
